Scheduling parallelizable tasks to minimize make-span and weighted response time

This paper presents a generalization to classical scheduling theory by removing the restriction that only one processor can work on a given task at a particular time. Instead, it is assumed that each task can be allocated any number of identical processors from one to the maximum number available, with each task's completion time being a function of the number of processors allocated. Tasks may be started any time, but once started, a task must not have its processor allocation altered or be preempted. Two objective functions are considered: minimizing the overall completion time for the tasks (make-span) and minimizing a weighted sum of the task completion times (weighted response). Both are considered subject to a constraint on the total number of processors available. Suboptimal algorithms are developed for both of these NP-hard problems using Lagrangian relaxation, and their performances are analyzed through extensive simulations. Duality gaps for all problems tested ranged from under 1% to 92%, depending more on the problem size than the specific problem.